Nonlinear dynamics of fronts

The global environmental imperative demands urgent actions on ecological stabilization, yet the global scale of such actions is persistently insufficient. This calls for investigating why the world economy appears to be so fearful of any potential environmental expenditure. Using the formalism of Lyapunov potential function it is shown that the stability principles for biomass in the ecosystem and for employment in economics are mathematically similar. The ecosystem has a stable and unstable stationary state with high (forest) and low (grasslands) biomass, respectively. In economics, there is a stable stationary state with high employment in mass production of conventional goods sold at low cost price, and an unstable stationary state with lower employment in production of novel products of technological progress sold at higher prices. An additional stable state is described for economics with very low employment in production of life essentials, such as energy and raw materials that are sold at greatly inflated prices. In this state the civilization pays 10% of global GDP for energy produced by a negligible minority of the working population (currently ∼0.2%) and sold at prices exceeding the cost price by 40 times, a state when any extra expenditures of whatever nature appear intolerable. The reason lies in the fundamental shortcoming of economic theory, which allows for economic ownership over energy sources. This is shown to be equivalent to equating measurable variables of different dimensions (stores and fluxes), which leads to effective violation of the laws of energy and matter conservation in modern economics.

Universite´Libre de Bruxelles, Service de Chimie-Physique, Case Postale 231, Boulevard du Triomphe, 1050 Bruxelles, Belgium~Received 28 May 1996!The numerical resolution of the Langevin equations, with specific internal noises deduced from masterequations, exhibits two qualitatively different behaviors for the reaction-diffusion wave fronts associated witheither a cubic or a quadratic chemical rate. In the case of a wave front between two stable stationary states,illustrated by the Schlo¨gl model, the effect of fluctuations in the vicinity of a bifurcation induces perturbativedeviations from the deterministic predictions on observable properties, like the propagation velocity, the profilewidth, and the value of the highest plateau. These deviations obey power laws that are determined. For wavefronts propagating into an unstable stationary state, such as in the Fisher model, a nonperturbative fluctuationeffect on velocity and profile width is observed, in relation to the selection, in the presence of noise, of aparticular solution in the continuum of linearly stable deterministic solutions. @S1063-651X~96!03010-3#PACS number~s!: 47.70.Fw, 82.20.2w, 05.40.1jI. INTRODUCTION

We study the dynamics of two charged point particles settling in a Stokes flow.We find what ranges of initial relative positions and what ranges of system parameters lead to formation of stable doublets.The system is parameterized by the ratio of radii, ratio of masses and the ratio of electrostatic to gravitational force.We focus on opposite charges.We find a new class of stationary states with the line of the particle centers inclined with respect to gravity and demonstrate that they are always locallyasymptotically stable. Stability properties of stationary states with the vertical line of the particle centers are also discussed.We find examples of systems with multiple stable stationary states.We show that the basin of attraction for each stable stationary state has infinite measure, so that particles can capture one another even when they are very distant, and even if their charge is very small. This behavior is qualitatively different from the uncharged case where there only exists a bounded set of periodic relative trajectories.We determine the range of ratios of Stokes velocities and ratio masses which give rise to non-overlapping stable stationary states (given the appropriate ratio of electrostatic to gravitational force). For non-overlapping stable inclined or vertical stationary states the larger particle is always above the smaller particle. The non-overlapping stable inclined stationary states existonly if the larger particle has greater Stokes velocity, but there are non-overlapping stable vertical stationary states where the larger particle has higher or lower Stokes velocity.

The unstable branch of stationary states in an illuminated thermochemical bistable system is stabilized with an external feedback loop which does not otherwise alter the autonomous system. The technique allows the measurement of the entire set of stationary states available to the system. Trajectories of transition between branches of stationary states (relaxation) are investigated theoretically and experimentally. Critical slowing down in the speed of the trajectories is predicted and measured as the points of marginal stability are approached. Trajectories are observed to spread as they depart from the unstable stationary state and then to coalesce as they approach a stable stationary state. These results are obtained due to experimental spread in initial conditions and not from noise along the relaxation trajectory.

The average rate of entropy production in a homogenous chemical system is investigated in oscillating periodic and chaotic modes as well as in coexisting stationary states. The simulations are based on an abstract model of a chemical reaction system with three freely varying concentrations. Five concentrations are assumed to be kept constant by suitable flows across the boundary. A fixed concentration is used as a control parameter. Second order mass action kinetics with reverse reaction is used. An unexpected result is that periodic modes in some windows in the chaotic interval have higher average rate of entropy production than the surrounding chaotic modes. A chaotic mode coexists with a stable stationary state with smaller entropy production. A unique (unstable) stationary state produces more entropy than the corresponding oscillating mode.

General principles of the theory of stability of solutions to differential equations are considered. The stability of equations describing the dynamics of changes in reagent concentrations in polyenzymatic biochemical chains is analyzed. Various mechanisms of formation of stable and unstable stationary states are considered, and unbalanced regimes and collapse are analyzed. The influence of systems of toxins and drugs on stability is studied. An interpretation of pathological processes based on stability theory is given.

In this paper the propagation of fronts into an unstable state are studied. Such fronts can occur e.g., in the form of domain walls in liquid crystals, or when the dynamics of a system which is suddenly quenched into an unstable state is dominated by domain walls moving in from the boundary. It was emphasized recently by Dee et al. that for sufficiently localized initial conditions the velocity of such fronts often approaches the velocity corresponding to the marginal stability point, the point at which the stability of a front profile moving with a constant speed changes. I show here when and why this happens, and advocate the marginal stability approach as a simple way to calculate the front velocity explicitly in the relevant cases. I sketch the physics underlying this dynamical mechanism with analogies and, building on recent work by Shraiman and Bensimon, show how an equation for the local ``wave number'' that may be viewed as a generalization of the Burgers equation, drives the front velocity to the marginal stability value. This happens provided the steady-state solutions lose stability because the group velocity for perturbations becomes larger than the envelope velocity of the front.For a given equation, our approach allows one to check explicitly that the marginal stability fixed point is attractive, and this is done for the amplitude equation and the Swift-Hohenberg equation. I also analyze an extension of the Fisher-Kolmogorov equation, obtained by adding a stabilizing fourth-order derivative -\ensuremath{\gamma} ${\ensuremath{\partial}}^{4}$\ensuremath{\varphi}/\ensuremath{\partial}${x}^{4}$ to it. I predict that for \ensuremath{\gamma}(1/12 the fronts in this equation are of the same type as those occurring in the Fisher-Kolmogorov equation, i.e., localized initial conditions develop into a uniformly translating front solution of the form \ensuremath{\varphi}(x-vt) that propagates with the marginal stability velocity. For \ensuremath{\gamma}>(1/12, localized initial conditions may develop into fronts propagating at the marginal stability velocity, but such front solutions cannot be uniformly translating. Differences between the propagation of uniformly translating fronts \ensuremath{\varphi}(x-vt) and envelope fronts are pointed out, and a number of open problems, some of which could be studied numerically, are also discussed.

In the last few decades, the stabilization in stationary states has emerged as a new auspicious campaigner in chaos theory and found a celebrated place through various control techniques such as predictive control, delayed feedback control, constant proportional feedback control and oscillating feedback control system. Generally, it is accepted that the superiority of control systems is not only to quash the irregular distribution of stationary states, but also to illustrate its basin of attractions as large as possible depending on the numerical as well as analytical observance. In this article, the universal stabilization in unstable stationary states is studied through superior fixed point feedback control system for a family of one-dimensional maps. Further, it is interesting to know that the novel system provides freedom in the control parameter γ due to which the stabilization increases more rapidly for the lager range of parameter γ in [0,1]. The analytical as well as numerical simulations are demonstrated to examine the behavior of parameter γ for which the unstable stationary state admits universal stability.

The eikonal approximation (instanton technique) is applied to the problem of large fluctuations of the number of species in spatially homogeneous chemical reactions with the probability density distribution described by a master equation. For both autocatalytic and nonautocatalytic reactions, the analysis of the distribution about a stable stationary state and of the transitions between coexisting stable states comes, to logarithmic accuracy, to the analysis of Hamiltonian dynamics of an auxiliary dynamical system. The latter can be done explicitly in a few cases, including one-species systems, systems with detailed balance, and systems close to the bifurcation points where the number of the stable states changes. In the last case, the fluctuations display universal features, and, for saddle-node bifurcation points, the logarithm of the probability of escape from the metastable state (per unit time) is proportional to the distance to the bifurcation point (in the parameter space) raised to the power 3/2. We compare the eikonal approximation for the stationary distribution of a master equation to Monte Carlo numerical solutions for two chemical two-variable systems with multiple stationary states, where none of the cited restrictions exists. For one of the systems in the pattern of optimal paths we observe caustics emanating from the saddle point.

A model based on five independent composition variables has been tested by simulating the complex bursting patterns observed experimentally for a Belousov–Zhabotinsky system in a flow reactor. If the system is in a region of constraint space where all stationary states are locally unstable, and if malonic acid concentration in the feed is decreased at constant flow of bromide ion, simple limit-cycle oscillations persist until the system reaches the Hopf bifurcation which generates a stable stationary state with low concentration of bromide ion. If bromide ion concentration in the feed is increased at constant flow of malonic acid, more and more small-amplitude oscillations are interspersed between trains of fewer and fewer large-amplitude excursions until a Hopf bifurcation leads to a stable stationary state with a high concentration of bromide ion. Patterns may be quite complex but usually repeat regularly. If a firing number is defined as ΣS/(ΣS+ΣL), where the summations represent the numbers of small- and large-amplitude oscillations in a complete pattern, then this firing number increases monotonically with increasing bromide ion in the feed. Narrow regions of chaotic mixing of patterns are also observed. The computations agree spectacularly with the careful experimental observations of Maselko and Swinney. Two separate curves are generated by the Hopf bifurcations which limit local stabilities of stationary states containing low and high concentrations of bromide ion, and the intersection of these curves provides a rationale for the ‘‘cross-shaped phase diagrams’’ which define regions of bistability and of oscillations in constraint spaces for such systems.

Results concerning the middle-latitude atmospheric flow in barotropic and quasi-geostrophic approximations are presented. With no dissipation, two semi-infinite regions of stability bound a finite region of instability, ensuring that the solutions are limited. In the presence of dissipation, above a critical value of forcing below which there is one stationary stable state, there are three stationary states, one unstable and two stable. Numerically we find that the two basins of attraction in phase space are strongly asymmetric, the one, corresponding to the solution resonant with the mountain, being much larger than the other. The model used for this study is the Charney-DeVore one.

The laser as nonlinear element for an optical associative memory

As is well known, the fluctuations from a stable stationary nonequilibrium\nstate are described by a linearized nonhomogeneous Boltzmann-Langevin equation.\nThe stationary state itself may be described by a nonlinear Boltzmann equation.\nThe ways of its linearization sometimes seem to be not unique. We argue that\nthere is actually a unique way to obtain a linear equation for the\nfluctuations. In the present paper we treat as an example an analytical theory\nof nonequilibrium shot noise in a diffusive conductor under the space charge\nlimited regime. Our approach is compared with that of Schomerus, Mishchenko and\nBeenakker [Phys. Rev. B 60, 5839 (1999)]. We find some difference between the\npresent theory and the approach of their paper and discuss a possible origin of\nthe difference. We believe that it is related to the fundamentals of the theory\nof fluctuation phenomena in a nonequilibrium electron gas.\n

Traditional cellular neural networks are arrays of coupled processing cells, where every uncoupled cell has two stationary stable states. In this paper an oscillatory function is used to define a cellular neural network whose uncoupled cells have a number of stable stationary states larger than two, so the output image has three or more grey levels. The proposed dynamics is applied to the decomposition of images with multiple gray levels.

In probabilistic reversal learning, the choice option yielding reward with higher probability switches at a random trial. To perform optimally in this task, one has to accumulate evidence across trials to infer the probability that a reversal has occurred. We investigated how this reversal probability is represented in cortical neurons by analyzing the neural activity in the prefrontal cortex of monkeys and recurrent neural networks trained on the task. We found that in a neural subspace encoding reversal probability, its activity represented integration of reward outcomes as in a line attractor model. The reversal probability activity at the start of a trial was stationary, stable and consistent with the attractor dynamics. However, during the trial, the activity was associated with task-related behavior and became non-stationary, thus deviating from the line attractor. Fitting a predictive model to neural data showed that the stationary state at the trial start serves as an initial condition for launching the non-stationary activity. This suggested an extension of the line attractor model with behavior-induced non-stationary dynamics. The non-stationary trajectories were separable indicating that they can represent distinct probabilistic values. Perturbing the reversal probability activity in the recurrent neural networks biased choice outcomes demonstrating its functional significance. In sum, our results show that cortical networks encode reversal probability in stable stationary state at the start of a trial and utilize it to initiate non-stationary dynamics that accommodates task-related behavior while maintaining the reversal information.